Melt the 4 tablespoons of butter in a heavy saucepan over medium heat. Add the onion and sauté for about 3 minutes. Add the mushrooms and sauté, stirring often, until they soften and wilt.

Sprinkle the flour into the mixture and continue to cook, stirring constantly to coat the onions and mushrooms. Add the broth gradually, stirring constantly, and bring to a full boil. Continue to boil for about 1 minute, stirring constantly. Reduce the heat and simmer for about 5 minutes.

Add the Half and Half, salt and pepper, and continue to cook over low. The mixture should remain just below a simmer.

Continue cooking for about ten minutes. Add the Madeira, stir thoroughly to combine, and serve sprinkled with snipped fresh parsley.